490 
ME. A. COHEN ON THE DIEEEEENTIAL COEFFICIENTS 
Moreover we know that the components of det (O, V,) are 
dz dy . dy 
dx dz (dx . dz 
Tt [it ^ 
dy dx dy 
Tt ‘^^~Tt COS X 
Let then the force P, which, besides gravity, acts on the particle, have for its 
components X, Y, Z, then, as Fj, the relative acceleration, has for its components 
d^^oc d^y d^z 
T^' ’df' evidently follows from equation (II.) of the preceding section, by revolving 
along the axes, that 
(px X 
' m 
Y 
d'^y Y /dz dx \ 
d^zZ dy 
(III.) 
d/CC d\t dz 
On multiplying these equations by ^ respectively, and adding, we find 
dx d’^x dy dPy dz dPz Xdx Ydy Zdz dz 
dt dP ' dt dp dt dP m di' m dt' m dt'^ dt 
This equation may also at once be deduced from the formula (II.) if we resolve 
along the direction of the particle’s relative motion, and observe that det (O, VJ is per- 
pendicular on that direction which coincides with the direction of Vj. 
By integrating the last equation, we see that the equation of vis viva applies to the 
particle’s relative motion just as if the particle’s relative motion were its actual motion, 
•with this difiierence only, that for the actual force of gravity the apparent force of gravity 
must be substituted. 
If the particle be a free particle acted on by no forces but gravity, then X=0, Y=0, 
Z=0, and the equations (III.) are linear, and are therefore easily integrated. 
Moreover if v, be the relative velocity, the equation of vis viva gives 
v\zzz2g{2 — /i), Ji being a constant. 
46. If the particle be suspended by a string from a point fixed to the earth, then if 
that point be taken as the origin, and X, Y, Z be the components of the string’s tension, 
we e'vddently have Z?/— Y2=0, X.z — Z.3:'=0, Yx — Xy=0 ; and substituting those values in 
the preceding equations (III.), we shall obtain two independent equations, which, together 
with the equation constant, will determine the particle’s relative motion. 
